Narrow Results

In this work, two originally separate adjustments for the Friedmann equations are concurrently considered. First, the fractal structure of the black hole horizon region is imposed by the Barrow entropy. The second adjustment is the $f(\mathcal{𝒯})$ gravity, which is based on a teleparallel framework generalization of the Einstein–Hilbert action, where $\mathcal{𝒯}$ is the scalar torsion. This can be considered as a Barrow entropy modification of the $f(\mathcal{𝒯})$ gravity thus yielding a new model. Gravity thermodynamics hypothesis principles are used to integrate these two models under a unified framework. We derive the modified Friedmann equation and note the corrections obtained. To understand the implications of such dual modifications, an application is analyzed and a particular $f(\mathcal{𝒯})$ toy model is chosen for the purpose. The equation of the state parameter of the resulting model, the dimensionless density parameters of matter and dark energy, and the deceleration parameter are explored to check the viability of the new model. A discussion of the dynamic evolution of the universe follows from these results. It is seen that the results comply with the observations. The newly developed model is promising and demands further study.

We study energy distribution in the context of teleparallel theory of gravity, due to matter and fields including gravitation, of the universe based on the plane-wave Bianchi VII_δ spacetimes described by the Lukash metric. For this calculation, we consider the teleparallel gravity analogs of the energy–momentum formulations of Einstein, Bergmann–Thomson and Landau–Lifshitz. We find that Einstein and Bergmann–Thomson prescriptions agree with each other and give the same results for the energy distribution in a given spacetime, but the Landau–Lifshitz complex does not. Energy density turns out to be nonvanishing in all of these prescriptions. It is interesting to mention that the results can be reduced to the already available results for the Milne universe when we write ω = 1 and Ξ² = 1 in the metric of the Lukash spacetime, and for this special case, we get the same relation among the energy–momentum formulations of Einstein, Bergmann–Thomson and Landau–Lifshitz as obtained for the Lukash spacetime. Furthermore, our results support the hypothesis by Cooperstock that the energy is confined to the region of nonvanishing energy–momentum tensor of matter and all non-gravitational fields, and also sustain the importance of the energy–momentum definitions in the evaluation of the energy distribution associated with a given spacetime.

We consider the energy–momentum definition of the Møller in both general relativity and teleparallel gravity to evaluate the energy distribution (due to both matter and fields including gravitation) associated with the topological black holes with a conformally coupled scalar field. Our results show that the energy depends on the mass M and charge Q of the black holes and cosmological constant Λ. In some special limits, the expression of the energy reduces to the energy of the well-known spacetimes. The results also support the viewpoint of Lessner that the Møller energy–momentum formulation is a powerful concept of the energy–momentum. Furthermore, the energy obtained in teleparallel gravity is also independent of the teleparallel dimensionless coupling constants which means that it is valid not only in the teleparallel equivalent of the general relativity but also in any teleparallel model.

The conserved quantities, that are, gravitational energy–momentum and its relevant quantities are investigated for cylindrical and spherical gravitational waves in the framework of teleparallel equivalent of General Relativity using the Hamiltonian approach. For both cylindrical and spherical gravitational waves, we obtain definite energy and constant momentum. The constant momentum shows consistency with the results available in General Relativity and teleparallel gravity. The angular momentum for cylindrical and spherical gravitational waves also turn out to be constant. Further, we evaluate their gravitational energy–momentum fluxes and gravitational pressure.

In this paper, we discuss the energy–momentum problem in the realm of teleparallel gravity. The energy–momentum distribution for a class of regular black holes coupled with a nonlinear electrodynamics source is investigated by using Hamiltonian approach of teleparallel theory. The generalized regular black hole contains two specific parameters α and β (a sort of dipole and quadrupole of nonlinear source) on which the energy distribution depends. It is interesting to mention here that our results exactly coincide with different energy–momentum prescriptions in general relativity.

In this paper, we work with teleparallel gravity and show that the values of the torsion vector and the torsion axial-vector depend on the choice of the tetrad field. We choose two different sets of tetrad fields, each of them satisfies the tetrad's conditions and leads to the Van Stockum spacetime. We obtain expressions for the torsion vector and the torsion axial-vector. In addition, we found that the obtained expressions are quite different for the two sets of tetrad fields. For the spacetime under consideration, if the metric coefficients are assumed to be functions of t only, it is then shown that the torsion axial-vector vanishes completely and the torsion vector still depends on the choice of tetrad fields. Finally, we found the spin precession of a Dirac particle in the spacetime under consideration.

We study the f(T) theory as an extension of teleparallel gravity and consider the Noether symmetry of Kantowski–Sachs (KS) anisotropic model for this theory. We specify the explicit teleparallel form of f(T) and find the corresponding exact cosmological solutions under the assumption that the Lagrangian admits the Noether symmetry. It is found that the universe experiences a power law expansion for the scale factors in the context of f(T) theory. By deriving equation of state (EOS) parameter, we show that the universe passes through the phantom and $\mathrm{\Lambda }$CDM theoretical scenarios. In this way, we estimate a lower limit age for the universe in excellent agreement with the value reported from recent observations. When KS model reduces to the flat Friedmann–Robertson–Walker (FRW) metric, our results are properly transformed into the corresponding values.

The teleparallel gravitational energy–momentum tensor density of the Friedmann–Lemaître–Robertson–Walker spacetime is calculated and analyzed: it is decomposed into density, isotropic pressure, non-isotropic pressures, and the heat-flux 4-vector; the antisymmetric part is decomposed into “electric” and “magnetic” components. It is found that the gravitational field obeys a radiation-like equation of state, the antisymmetric part does not contribute to the gravitational energy–momentum; and the total energy density, the non-isotropic pressures and the heat-flux 4-vector vanish for spatially flat universes. Finally, it is verified that the field equations have a well-defined vacuum.

In this paper, we discuss the dynamics of the anisotropic Bianchi I Universe filled with perfect fluid in teleparallel $f(T)$-gravity. Using both analytical and numerical approaches, we confirm the main results of the previous authors such as the presence of a future isotropic attractor, and find a number of new ones such as the existence of an essential number of bounce solutions and the possibility to construct late-time acceleration without any specific matter.

In the framework of general relativity (GR) and teleparallel equivalent to GR (TEGR), we investigate the energy–momentum localization of a stringy charged black hole by using Bergman–Thompson’s prescription and the Papapetrou’s prescription. Moreover, we investigate the gravitational energy–momentum (GEM) in TEGR. Our results are consistent with the results of previous studies. Finally, we calculate the GEM flux and gravitational center of mass in TEGR.

The energy distribution in the Locally Rotationally Symmetric (LRS) Bianchi type II space–time is obtained by considering the Møller energy–momentum definition in both Einstein's theory of general relativity and teleparallel theory of relativity. The energy distribution which includes both the matter and gravitational field is found to be zero in both of these different gravitation theories. This result agrees with previous works of Cooperstock and Israelit, Rosen, Johri et al., Banerjee and Sen, Vargas, and Aydogdu and Salti. Our result — the total energy of the universe is zero — supports the view points of Albrow and Tryon.

We calculate the total energy of an exact spherically symmetric solutions, i.e. Schwarzschild and Reissner–Nordström, using the gravitational energy–momentum 3-form within the tetrad formulation of general relativity. We explain how the effect of the inertial makes the total energy unphysical. Therefore, we use the covariant teleparallel approach which makes the energy always physical one. We also show that the inertial has no effect on the calculation of momentum.

Inspired by the Bañados, Teitelboim and Zanelli (BTZ) formalism, we discuss the Maxwell-$f(T)$ gravity in $(2+1)$ dimensions. The main task is to derive exact solutions for a special form of $f(T)=T+𝜖T^2$, with $T$ being the torsion scalar of Weitzenböck geometry. To this end, a triad field is applied to the equations of motion of charged $f(T)$ and sets of circularly symmetric noncharged and charged solutions have been derived. We show that, in the charged case, the monopole-like and the $ln$ terms are linked by a correlative constant despite the known results in teleparallel geometry and its extensions.${}^{39}$ Furthermore, it is possible to show that the event horizon is not identical with the Cauchy horizon due to such a constant. The singularities and the horizons of these black holes are examined: they are new and have no analogue in the literature due to the fact that their curvature singularities are soft. We calculate the energy content of these solutions by using the general vector form of the energy–momentum within the framework of $f(T)$ gravity. Finally, some thermodynamical quantities, like entropy and Hawking temperature, are derived.

Using the Einstein, Bergmann–Thomson (BT) and Landau–Lifshitz (LL) energy and momentum formulations in teleparallel gravity (TG), we obtain the total energy and momentum distributions for phantom black hole metric. We get different energy distributions similar to the earlier study and the momentum distributions vanish for phantom black hole metric in TG. These momentum solutions agree with the study of Sahoo et al. in general relativity. However, using Einstein, Bergmann–Thomson and Landau–Lifshitz energy and momentum complexes, we investigate regular black hole, asymptotically flat wormhole, anti-de Sitter wormhole and Ellis wormhole solutions in TG. We obtain (i) same BT and LL energy density solutions for regular black hole metric, (ii) same and zero energy distribution for asymptotically flat wormhole, (iii) proportion with Einstein and BT energy density solutions for anti-de Sitter wormhole, (iv) same and negative Einstein and BT energy density solutions for Ellis wormhole in TG.

In this study, we obtain Einstein, Bergmann–Thomson (BT), Landau–Lifshitz (LL), Møller, Papapetrou (PP) and Tolman energy–momentum (EM) distributions for Ruban universe model in general relativity (GR) and teleparallel gravity (TG). We obtain same results for Einstein, Bergmann–Thomson and Landau–Lifshitz energy–momentum distributions in GR and TG. Also, we get same results for Einstein and Tolman energy–momentum distributions in GR. The Møller energy–momentum results are different in GR and TG. Also, using Ruban universe model, we obtain LRS Bianchi type I solutions and we get zero energy–momentum results for this universe model in GR and TG. These results of LRS Bianchi type I universe model agree with Aygün et al., Taşer et al., Doğru et al., Banerjee–Sen, Tryon and Xulu in different gravitation theories.

This paper investigates the cosmological dynamics of the Hessence scalar field coupled with the dark matter in the background of the teleparallel gravity. We have assumed that the potential of the scalar field is exponential in nature whereas the $f(T)$ appearing in the teleparallel theory has the form $f(T)=\beta {(-T)}^{\frac{1}{2}}$. The field equations of this system reduce to a nonlinear autonomous system and dynamical system analysis is then performed. Due to the nonlinearity and the existence of multiple zero eigenvalues, the traditional procedures of analysis break down. So some novel technique is required. One of the latest such techniques is the invariant manifold theory. By the application of this theory, one projects the variables linked with the zero eigenvalues onto the variables linked with the nonzero eigenvalues to compute the center manifolds and the reduced systems associated with the critical points. These reduced systems reflect the nature of the whole dynamical systems. They also have less dimension and are often simple in nature. Hence, it is possible to solve them directly. In this paper, we work exactly in this spirit and find the center manifolds and solve the corresponding reduced system for some of the critical points associated with the dynamical system. We discover some interesting results namely that there are certain bounds on the interaction term $\delta$ which asserts the stability of the systems. We also present various stability diagrams of the reduced systems. An asymptotic analysis is then done for the critical points at infinity. Finally, we discuss the cosmological interpretation of our results.

In a recent work we derived the kinematic Hamiltonian and primary constraints of the new general relativity class of teleparallel gravity theories and showed that these theories can be grouped in 9 classes, based on the presence or absence of primary constraints in their Hamiltonian. Here we demonstrate an alternative approach towards this result, by using differential forms instead of tensor components throughout the calculation. We prove that also this alternative derivation yields the same results and show how they are related to each other.

The existence of an extra degree of freedom (d.o.f.) in $f(T)$ gravity has been recently proved by means of the Dirac formalism for constrained Hamiltonian systems. We will show a toy model displaying the essential feature of $f(T)$ gravity, which is the pseudo-invariance of $T$ under a local symmetry, to understand the nature of the extra d.o.f.

The teleparallel gravity, in the weak field approximation, is considered. In the Schwarzschild space–time the gravitational Stefan–Boltzmann law is calculated. Using the first law of thermodynamics, the Hawking entropy and temperature of the black hole are calculated. Thermofield dynamics formalism is used to define the temperature-dependent theory. In addition the gravitational Casimir effect and Stefan–Boltzmann law at zero and finite temperature are calculated in Schwarzschild space–time.

The role played by torsion in gravitation is critically reviewed. After a description of the problems and controversies involving the physics of torsion, a comprehensive presentation of the teleparallel equivalent of general relativity is made. According to this theory, curvature and torsion are alternative ways of describing the gravitational field, and consequently related to the same degrees of freedom of gravity. However, more general gravity theories, like for example Einstein–Cartan and gauge theories for the Poincaré and the affine groups, consider curvature and torsion as representing independent degrees of freedom. By using an active version of the strong equivalence principle, a possible solution to this conceptual question is reviewed. This solution ultimately favors the teleparallel point of view, and consequently the completeness of general relativity. A discussion of the consequences for gravitation is presented.